Question

A card is drawn, with replacement, from a regular deck of cards 16 times. Let random variable X represent number of clubs among those 16 cards selected (there are 13 clubs in every deck; there are 52 cards in a deck). Find the variance of X,

Answer 1

Answer:

$Var(X)=3$

Step-by-step explanation:

1) Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

The probability mass function for the Binomial distribution is given as:

$P(X)=(nCx)(p)^x (1-p)^{n-x}$

Where (nCx) means combinatory and it's given by this formula:

$nCx=\frac{n!}{(n-x)! x!}$

And the expected value is given by $E(X)=np$ and the variance $Var(X)=np(1-p)$

2) Solution to the problem

First we need to define our random variable $X$ represent the number of clubs amont 16 cards selected from a standard deck of 52 cards.

Since the extraction it's with replacment for each trial of this experiment the probability of getting a club card is $p_{club}=\frac{13}{52}$, because there are 13 club cards in a standard deck of 52 cards. The number of times that the experiment would be repeated are $n=16$ times.

So then our random variable X follows this distribution $X \sim Binom(n=16,p=\frac{13}{52})$

So the possible values for the random variable are X=0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16. Since the experiment is with replacement, and on this case on each trial we have the same probability of get a club card.

The expected value for the Bonimial distribution is given by:

$E(X)=np=16*\frac{13}{52}=4$

And the variance by:

$Var(X)=np(1-p)=16*\frac{13}{52}(1-\frac{13}{52})=3$